AI 中文总结
本文提出加速Fourier-Motzkin消元的方法，包括安全结合冗余移除与线性规划，以及基于资源使用的变量消元顺序规则，显著降低运行时间和中间不等式数量。
AI 中文摘要
Fourier-Motzkin消元通过每次消去一个变量来计算多面体在其坐标子集上的投影的不等式描述。它被用于优化和计算机科学的多个领域，并且是获得因果结构的熵约束的标准方法，其中对潜在变量的边缘化产生这样的投影。其局限性在于中间不等式系统的增长，即使投影本身仅以单指数增长，中间系统也可能以双指数增长。在实践中，该方法的计算负担因此取决于两个选择：每一步后如何移除冗余不等式，以及变量消元的顺序。我们考虑这两者。我们首先通过一个显式例子表明，Imbert的冗余测试不能与通过线性规划进行的冗余移除交错进行。然而，我们证明，如果Imbert测试使用的推导记录在使用线性规划的每一步后被重新初始化，这两种方法可以可靠地结合。然后，我们提出一个选择变量消元顺序的规则，该规则提供了显著的计算优势，但代价是资源使用增加。我们在一些随机多面体上展示了这一优势，在该规则下，与固定顺序下的相同消元相比，运行时间减少了6到25倍。对于因果结构的熵描述，当有超过250个不等式和超过100个变量需要消去时，我们的规则使每一步处理的不等式数量比固定顺序低一到两个数量级。
英文摘要
Fourier-Motzkin elimination computes an inequality description of the projection of a polyhedron onto a subset of its coordinates by eliminating one variable at a time. It is used in several areas of optimisation and computer science, and it is a standard way of obtaining the entropic constraints of a causal structure, where the marginalisation over the latent variables produces such a projection. Its limitation is the growth of the intermediate systems of inequalities, which can be doubly exponential in the number of eliminated variables even though the projection itself grows only as a single exponential. In practice the computational overload of the method therefore depends on two choices: how the redundant inequalities are removed after each step, and the order in which the variables are eliminated. We consider both. We first show, by an explicit example, that Imbert's redundancy test cannot be interleaved with redundancy removal by linear programming. We show that the two methods, however, can be combined soundly if the derivation records used by Imbert's test are re-initialised after every step at which linear programming is used. We then propose a rule for choosing the elimination order of the variables that gives a significant computational advantage, however, at the cost of increased resource usage. We demonstrate this advantage on some random polytopes, where the rule reduces the running time by factors of between 6 and 25 compared with the same elimination under a fixed order. For entropic descriptions of causal structures, with more than 250 inequalities and more than 100 variables to eliminate, our rule keeps the number of inequalities handled at each step one to two orders of magnitude lower than a fixed order.
Comments19 pages, 6 Figures. Preliminary version, some results will be added
